Frame rate drops in video editor
Why Is It Slow? 

1. The V.S.E. , unlike your typical video editor, is frame dependent. Most others rely on time. This is why you must set the correct fps in the properties panel. Which you did, great! 
Second Blender has to "cache" those frames. So when you play it through it takes each frame and saves a temporary version of it. But, the farther you go the more it will struggle to cache it, meaning you may have to play through it a couple times for it to work.
OR
2. You've reached the maximum cache... 

Ok... So how do I fix that?

To change max cache press CTRL+ALT+U to bring up user preferences. Head over to the last tab labeled System and scroll down to the bottom where you'll see Sequencer/Clip:Editor. Go ahead and set that to your RAM. You can go even higher if you want; I can't guarantee what will happen. Make sure to save your user settings. 

Other Things You Can Try: 
1. Use that handy-dandy proxy setting. Right click on your video, and set that proxy! Then you'll have to go down and press Rebuild Proxy and Timecode Indices
Use frame dropping or Av-sync. Use it, it does what it says. Drops frames to run faster. 

Last Thing 

Not entirely sure why it says it runs at 60 fps (its not), but I suppose its just saying hey, this is how fast I can run. 
Also know that this will NOT affect the final render in any way, only in the viewport.